We report $^{77}$Se-nuclear magnetic resonance (NMR) results down to sufficiently low temperatures under magnetic fields parallel to both the $ab$-plane and the c-axis in a paramagnetic/superconducting (PM/SC) phase of K$_x$Fe$_{2-y}$Se$_2$. The observation of anisotropy in the orbital part of the Knight shift results in the anisotropy of its spin part increasing on approaching the transition temperature. The anisotropy of the Korringa relation suggests the presence of the weak spin fluctuations with a finite wave vector $bm{q}$, which induce the magnetic fluctuations along the ab-plane at the Se site. Such fluctuations do not correspond to the stripe $(pi,0)$ correlation of the Fe moment observed in many Fe-based superconductors, and are not contradictory to weak $(pi,pi)$ correlations. The nuclear spin-lattice relaxation rate $1/T_1$ shows a field-independent $T_1T sim const.$ behavior at low temperatures for $H parallel ab$, which is attributed to the nonzero density of states at the Fermi level and can be explained by the sign-changing order parameter even for nodeless gaps. The temperature dependence of $1/T_1$ is reproduced well by nodeless models with two isotropic gaps or a single anisotropic gap. The obtained gap magnitude in the isotropic two-gap model is comparable to those obtained in the angle-resolved photoemission spectroscopy experiments.
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